Proposition. If a relation ≺ on a set X is well-founded, extensional and transitive then it is a strict total order, thus defining a well-order.
Proof. Denote x≼y ⇔ (x≺y ∨ x=y) and ∀x∈X, Bx = {y∈X | x≼y ∨ y≺x}.
The restriction of a well-order to any subset is also a
well-order. So any subset of ℕ is well-ordered. In fact the
well-orders on finite sets are the total orders, and they are
isomorphic as soon as they have the same number of elements.
But there are well-orders bigger than ℕ too. The description of
any well-order is always the same: it starts from the smallest
element (0), then the smallest above 0 (1), then the smallest
above 1 (2), and so on up to infinity; then the smallest after all
natural numbers, then the next one, then the next,... and so on to
infinity, then the next... and so on and so forth, even possibly
beyond any description, but this can stop at any time. See Chaitin's
speech for a more explicit description.
An initial segment of a well-ordered set E is a subset of E of
the form {x∈E | x<y} for some y∈E..
A fundamental property is that for any two well-ordered sets
E and F, there exists a unique isomorphism that is either between
E and F, or between E and an initial segment of F, or between an
initial segment of E and F.
An ordinal can be defined as an isomorphism class of
well-ordered sets. The traditional study of ordinals, and thus of
well-orders, formalizes each ordinal by its only representative
whose strict well-order is given by the membership predicate (
∈ ) and that are transitive (x∈A∈E ⇒ x∈E). In such conditions we
have {x∈E | x<y}=y, which may simplify things. But the proof of existence of
such an ordinal isomorphic to any well-ordered set, requires the
schema of replacement to be proven. Here we shall call ZF-ordinals
the transitive sets well-ordered by ∈.
In fact most properties of well-orders and their isomorphism
classes can as well be developed without such a representative,
thus without the axiom schema of replacement, though it may be a
little bit more tedious.
Any set of ordinals (as classes of well-ordered sets) is well-ordered by defining "equality" by the isomorphism, and the order between non-isomorphic well-orders is given by which of both sets is isomorphic to an initial segment of the other.
In the case of ZF-ordinals this all simplifies as the same
predicate ∈ defines the strict well-order inside and between
all ordinals.
For any ordinal there is a bigger one obtained by adding an
element after it. In fact, for any well-ordered set E, we can
define the "next" well-ordered set as the set of all initial
segments of E plus E itself. That is E ∪{E} in the case of
ZF-ordinals. And for any set of ordinals we can take their
"union", that is the smallest ordinal that is larger than them
all. This is simply the union as sets in the case of ZF-ordinals.
Among all ordinals in bijection with a given ordinal, there is a
smallest one. Such a smallest ordinal inside a cardinal (an
ordinal not in bijection with any initial segment of itself) will
be chosen to represent this cardinal.
If we could well-order any set (thus, if we accept the axiom of
choice) then we could simply argue that the class of
cardinal-ordinals does not form a set because this has been shown
for cardinals in general.
For example, ℕ is not the biggest cardinal because P(ℕ) is bigger
and any well-order on it gives an ordinal with cardinality
larger than ℕ.
Otherwise the traditional method also uses the axiom schema of
replacement, making it unclear which assumptions are needed to
show that the class of cardinal-ordinals is not a set. However we
can still deduce it from just the powerset axiom, without either
the axiom of choice nor the axioms of replacement, but this is a
little bit more subtle. Here it goes:
For any well-ordered set X, take the set A={R⊂X×X | R is a
well-order of a subset of X}. Then the quotient of A by the
relation of isomorphism, is naturally a well-ordered set whose
ordinal is the smallest ordinal with cardinality strictly larger
than X. Because if it was in bijection with a subset of X then it
would define a well-order of this subset...
This is the smallest uncountable cardinal-ordinal, thus a
candidate as an intermediate cardinal between ℕ and P(ℕ).
According to the above, it can be explicitly constructed out of
P(ℕ), as the quotient of a well-defined subset of P(ℕ) ≈
P(ℕ×ℕ) by a well-defined equivalence relation. Thus there
exists a surjection from P(ℕ) to it.
Using the axiom of choice it can be represented as a subset of
P(ℕ).
The continuum hypothesis takes the form
of the statement "There is a bijection exists between Aleph1 and
P(ℕ)".